Universality conjecture for critical one-dimensional random Schrödinger operators

Let $E$ be distributed according to the arcsine law, $U$ be uniform on $[0,1]$, and let $\mathcal{Z}$ be a standard two-sided Brownian motion started from $0$, independent of $E$ and $U$. Define

$$\rho(E)=\frac{1}{\sqrt{1-E^2/4}}\mathbf{1}_{|E|<2},\qquad \tau(E)=(\sigma\rho(E))^2,$$

and

$$S(t)=\exp\left(\frac{\mathcal{Z}_t}{\sqrt{2}}-\frac{|t|}{4}\right).$$

Let $H_n$ be the finite one-dimensional random Schrödinger operator, choose an eigenvalue $\mu$ uniformly from its eigenvalues, and let $\psi^\mu$ be the corresponding normalized eigenvector. Universality conjecture. The scaling limit of the eigenvector picked from the spectral measure at zero of the infinite one-dimensional random Schrödinger operator is as described by

$$\left( \mu, n\left|\psi^\mu(\lfloor nt\rfloor)\right|^2dt\right)\Rightarrow\left(E,\frac{S(\tau(E)(t-U))dt}{\int_0^1ds\,S(\tau(E)(s-U))}\right).$$

This conjecture proposes universality of the critical eigenvector scaling limit beyond the discrete model studied in the paper, in the Poisson limit. The source does not state a resolution or provide further evidence of known cases, so its status remains open.